Just as functions of several variables may be differentiated with respect to two or more of them, so may their integrals with respect to more than one variable be formed. The formal definitions of such multiple integrals are extensions of that given for a single variable in Chapter 3. In this chapter, we first discuss double and triple integrals and illustrate some of their applications. We then consider how to change the variables in multiple integrals and, finally, discuss some general properties of Jacobians.
8.1 Double integrals
For an integral involving two variables – a double integral – we have a function, f(x, y) say, to be integrated with respect to x and y between certain limits. These limits can usually be represented ...